Ince equation
From HandWiki
In mathematics, the Ince equation, named for Edward Lindsay Ince, is the differential equation
- [math]\displaystyle{ w^{\prime\prime}+\xi\sin(2z)w^{\prime}+(\eta-p\xi\cos(2z))w=0. \, }[/math]
When p is a non-negative integer, it has polynomial solutions called Ince polynomials. In particular, when [math]\displaystyle{ p=1, \eta\pm\xi=1 }[/math], then it has a closed-form solution[1]
[math]\displaystyle{ w(z)=Ce^{-iz}(e^{2iz}\mp 1) }[/math]
where [math]\displaystyle{ C }[/math] is a constant.
See also
- Whittaker–Hill equation
- Ince–Gaussian beam
References
- ↑ Cheung, Tsz Yung. "Liouvillian solutions of Whittaker-Ince equation". Journal of Symbolic Computation 115 (March-April 2023): 18-38. doi:10.1016/j.jsc.2022.07.002.
- Boyer, C. P.; Kalnins, E. G.; Miller, W. Jr. (1975), "Lie theory and separation of variables. VII. The harmonic oscillator in elliptic coordinates and Ince polynomials", Journal of Mathematical Physics 16 (3): 512–517, doi:10.1063/1.522574, ISSN 0022-2488, Bibcode: 1975JMP....16..512B, http://researchcommons.waikato.ac.nz/bitstream/10289/1243/1/Kalnins%20variables%207.pdf
- Magnus, Wilhelm; Winkler, Stanley (1966), Hill's equation, Interscience Tracts in Pure and Applied Mathematics, No. 20, Interscience Publishers John Wiley & Sons\, New York-London-Sydney, ISBN 978-0-486-49565-1, https://books.google.com/books?id=ML5wm-T4RVQC
- Mennicken, Reinhard (1968), "On Ince's equation", Archive for Rational Mechanics and Analysis (Springer Berlin / Heidelberg) 29 (2): 144–160, doi:10.1007/BF00281363, ISSN 0003-9527, Bibcode: 1968ArRMA..29..144M
- Wolf, G. (2010), "Equations of Whittaker–Hill and Ince", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, http://dlmf.nist.gov/28.31
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